P Values Maximized Over a Confidence Set for the Nuisance Parameter

Abstract

Testing problems are often complicated by the presence of a nuisance parameter vector .Consider first a model in which there is no nuisance parameter.Suppose the data X have a probability distribution P defined in terms of a parameter , and we wish to test the simple hypothesis H 0 : = 0 .If the test statistic T is used to test H 0 and if large values of T give evidence against H 0 , then for an observed value T = t, the p-value is p = P 0 (T t).Now consider a model with a nuisance parameter .The distribution of X has two parameters, and .We still wish to test H 0 : = 0 , but this hypothesis is no longer simple, because the value of is unspecified.Using a test statistic as above, the p-value is now p = sup P 0 ; (T t). (See, for example, Bickel and Doksum (1977), pp.171-172).Unfortunately, the need to calculate the sup has complicated the problem.This complication is usually handled in one of three ways.First, in some problems it can be shown that, for all values of t, the sup is always attained at a particular value 0 .In this case the p-value is simply p = P 0 ; 0 (T t), and the parameter ( 0 ; 0 ) is called